Purinergic Receptors We have previously described (see 2010 report) a kinetic model, developed in collaboration with the Stojilkovic experimental lab (NICHD), of the P2X7 receptor. This receptor is a ligand-gated ion channel activated by extracellular ATP and is expressed ubiquitously, including in pituitary cells and macrophages. At low concentrations of ATP, this ligand-gated calcium channel acts much like other members of the P2X family, but prolonged or repeated exposure to high ATP concentrations, causes it to dilate and gate a massive influx of calcium. This unusual and complex behavior had led to proposals that the normal and super-normal currents are due to two different channels, but the model, a Markov state model with 8 states, showed that a single channel could play both roles. In neurons, P2X7 may act mostly as a conventional calcium channel, but in macrophages, the small current mode may promote cell growth and differentiation whereas the large current mode may lead to cell death. This could be an important part of how the immune system maintains a balance between responding appropriately and over-reacting to inflammation. A polymorphism in the P2X7 receptor has been proposed as a susceptibility gene for the NOD mouse, a model for type 1 diabetes. We have now extended the model to describe the rather different behavior of the P2X2 receptor (Ref. # 1). The current through this receptor rapidly shuts off in the face of maintained ATP but is nonetheless thought to dilate, because it gains the ability to conduct large organic cations after stimulation with ATP. The model showed how desensitization could occur simultaneously with dilation. With the help of the model, we were able to identify not one but two distinct mechanisms of desensitization, one calcium-dependent and one calcium-independent. Each of these adds 4 states to the Markov model for a total of 16. The model showed that the calcium-dependent desensitization must be mediated by an effector molecule that exhibits hysteresis in order to account for the observed kinetics. Although a kinetic model cannot by itself identify molecules, the predicted kinetics should help guide the search. The P2X2R model includes the P2X7R as a sub-model, albeit with modified rate constants. The similarity will be heightened by adding calcium-independent desensitization to the P2X7R model, a feature which was known to be present but was neglected in the first iteration in order to keep the focus on the key features of dilation and memory. These connections should help elucidate the evolutionary and developmental unity of the P2X family. In the coming period, we plan to push this approach further by modeling other members of the P2X family. Classification of Bursting Oscillations Bursting oscillations, which are widely found in neurons and endocrine cells, consist of silent and active (spiking) periods alternating on a time scale typically of seconds to tens of seconds. These are often mediated by the rise and fall of intracellular calcium, which modulates the spiking activity through calcium-activated potassium channels, but many other channel mechanisms are possible. Mathematically, we abstract from the channels involved to focus on the transitions, known as bifurcations, between spiking and silence. In addition to modeling the bursting electrical oscillations in particular cell types, we have had a long-standing interest in classifying the various burst mechanisms based on the types of these transitions. This makes it possible to establish the simplest models that can account for each of the known patterns, where simplicity is indexed by the minimum number of parameters required to define the set of transitions. In general, cells use redundant parametrizations because they have to explore parameter space through the mechanisms at their disposal, such as ion channels and pumps, but the minimal mathematical description is still important for our understanding. We can now report major progress, in collaboration with Krasimira Tsaneva-Atanosova of the University of Bristol and Hinke Osinga of the University of Auckland, that largely completes the program of analysis of bursting begun in this lab by former chief John Rinzel 25 years ago. See Ref. # 2. This work was also presented as an invited plenary talk at the 2012 Life Sciences meeting of the Society for Industrial and Applied Mathematics. One application of the classification scheme is to understand the connections between the bursting patterns exhibited by different cells types. A key motivating example was the comparison between pituitary somatotrophs and lactotrophs vs. pancreatic beta cells. Both classes of cells exhibit bursts of action potentials from a depolarized plateau, which is potent at driving calcium entry into the cells for secretion of their respective hormones. We had shown in previous work that in the pituitary models the spikes were transients due to slow attraction to the upper steady state that defines the plateau, whereas in the beta cells the spikes are sustained oscillations that would persist if calcium were fixed rather than slowly varying. The two types of cells are developmental and evolutionary cousins, so it was expected that the distinct burst mechanisms would be related. However, whereas it has been known for a while that the beta-cell type burst pattern can be generated with only three parameters, we found that the pituitary pattern requires four parameters, further underscoring the fundamental differences between the two. Nonetheless, both types appear within the extended four-parameter scheme as neighbors and each can be converted into the other varying only one of the parameters. We and others had found previously that each of several biophysically identifiable parameter, including the threshold for activation of the calcium channels, can mediate such a conversion. The abstract mathematics is thus gratifyingly confirmatory of the biophysics, but this raises other questions. If activity patterns lie nearby in parameter space, how are cells able to maintain their identities and not drift from one to another? We believe this is a deep question and we hope to be able to address it in the future. In addition to the specific projects described here, we were involved in organizing a highly successful workshop on modeling in neuroendocrinology, in Tours, France. This was the third semi-annual meeting on this topic involving both theorists and experimentalists, and the nascent interdisciplinary community that has grown out of the meetings now appears to be well-established and self-sustaining.